Local Search
Foundations of Constraint Processing
CSCE421/821, Spring 2009:
www.cse.unl.edu/~choueiry/S09-421-821/
Berthe Y. Choueiry (Shu-we-ri)
Avery Hall, Room 360
choueiry@cse.unl.edu
Tel: +1(402)472-5444
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
1
Lecture sources
Required reading
1. Dechter: Chapter 7, Section 7.1 and 7.2
Recommended
1. R. Bartak’s online guide:
http://kti.ms.mff.cuni.cz/~bartak/constraints/stochastic.html
1. AIMA:
Section 4.4 (1st edition) Section 4.3 (2nd edition)
1. Bresina 96:
Heuristic-Biased Stochastic Sampling.
AAAI/IAAI, Vol. 1 1996: 271-278
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
2
Solving CSPs
CSPs are typically solved with a combination of
1. Constraint propagation (inference)
2. Search (conditioning)
• Backtrack search
• Local search
We focus on local search
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
3
Outline
•
•
•
•
General principle
Main types: greedy & stochastic
When nothing works…
Evaluation methods
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
4
Backtrack search
•
Properties
– Systematic and exhaustive
– Deterministic or heuristic
– Sound and complete
•
Shortcomings
– worst-case time complexity prohibitive
– often unable to solve large problems. Thus, theoretical
soundness and completeness do not mean much in practice
•
Idea
– Use approximations: sacrifice soundness and/or completeness
– Can quickly solve very large problems (that have many solutions)
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
5
Local search: the picture
•
•
•
•
States are laid up on a surface
State quality/cost is its height
State space forms a landscape
Optimum state:
– maximizes solution quality
– minimizes solution cost
• Move around from state to state and try to reach
the optimum state
• Exploration restricted to neighboring states, thus
‘local’ search (ref. Holger & Hoos)
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
6
Components of a local search
• State
– is a complete assignment of values to variables, a possibly
inconsistent solution
• Possible moves
– are modifications to the current state, typically by changing the
value of a single variable. Thus, ‘local’ repair (ref. Dechter)
– Examples:
• SAT: Flipping the value of a Boolean variable (GSAT),
• CSPs: Min-conflict heuristic (variations)
• Evaluation (cost) function
– rates the quality of a state, typically in the number of violated
constraints
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
7
Generic Mechanism
• Cost function: number of broken constraints
• General principle
– Start with a full but arbitrary assignment of values to variables
– Reduce inconsistencies by local repairs (heuristic)
– Repeat until
• A solution is found
• The solution cannot be repaired
• … You run out of patience
(global minimum)
(local minimum)
(max-tries)
• A.k.a.
– Iterative repair (decision problems)
– Iterative improvement (optimization problems)
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
8
Outline
•
•
•
•
General principle
Main types: greedy & stochastic
When nothing works…
Evaluation methods
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
9
Main types of local search
1. Greedy:
– Use a heuristic to determine the best move
2. Stochastic (improvement)
– Sometimes (randomly) disobey the heuristic
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
10
Greedy local search
• At any given point,
– make the best decision you can given the
information you have and proceed.
– Typically, move to the state that minimizes the
number of broken constraints
• Example:
– hill climbing (a.k.a. gradient descent/ascent)
– Local beam search: keep track of k states
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
11
Greedy local search
• Problems:
– local optima (stuck),
– plateau (errant),
– ridge (oscillates from side to side, slow progress)
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
12
Stochastic Local Search
• Sometimes (randomly) move to a state that is
not the best: use randomization to escape local
optimum
• Examples:
• RandomWalk (stochastic noise)
• Tabu Search
 Simulated Annealing
 Generic algorithms
 Breakout method (constraint weighting)
 ERA (multi-agent search)
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
13
Simulated Annealing: idea
• Analogy to physics:
– Process of gradually cooling a liquid until it freezes
– If temperature is lowered sufficiently slowly, material
will attain lowest-energy configuration (perfect order)
• Basic idea:
– When stuck in a local optimum, allow few steps
towards less good neighbors to escape the local
maximum
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
14
Simulated Annealing: Mechanism
• Start from any state at random, start countdown and loop
until time is over:
– Pick up a neighbor at random
– Set d = quality of neighbor – quality of current state
– If d>0 (there is improvement)
• Then move to neighbor & restart countdown
• Else, move to neighbor with a transition probability p<1
• Transition probability proportional to ed/t
– d is negative, and t is time countdown
– As times passes, less and less likely to make the move towards
unattractive neighbors
• Under some very restrictive assumptions, guaranteed to
find optimum
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
15
Properties
• Non-systematic and non-exhaustive
• Liable to getting stuck in local optima (optima/minima)
• Non-deterministic:
– outcome may depend on where you start
• Typically, heavy tailed:
– probability of improving solution as time goes by quickly
becomes small but does not die out
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
16
Genetic Algorithms
• Basic step: Combinations two complete assignments
(parents) to generate offsprings
• Mechanism
– Starts from an initial population
– Encodes assignments in a compact manner (a string)
– Combines partial solutions to generate new solutions (next
generation)
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
17
Genetic Algorithm (2)
• Fitness Function ranks a state’s quality, assigns probability for
selection
• Selection randomly chooses pairs for combination depending on
fitness
• Crossover point randomly chosen for two individuals, offsprings are
generated
• Mutation randomly changes a state
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
18
Breakout strategies
[Bresina]
• Increase the weights of the broken
constraints so that they are less likely to
be broken in subsequent iterations
• Quite effective for recovering from local
optima
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
19
ERA: Environment, Rules, Agents
[Liu et al, AIJ 02]
•
•
•
•
•
Environment is an nxa board
Each variable is an agent
Each position on board is a value of a domain
Agent moves in a row on board
Each position records the number of violations
caused by the fact that agents are occupying
other positions
• Agents try to occupy positions where no
constraints are broken (zero position)
• Agents move according to reactive rules
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
20
Reactive rules
[Liu et al, AIJ 02]
Reactive rules:
– Least-move: choose a position with the min. violation value
– Better-move: choose a position with a smaller violation value
– Random-move: randomly choose a position
Combinations of these basic rules form different behaviors.
Reactive rules
Behavior
designer
LR
least-move with 1-p and random-move with p
BR
better-move with 1-p and random-move with p
BLR
first better-move, if fail then apply LR
rBLR
first apply better-move r times, if fail then apply LR
FrBLR
apply rBLR in the first r iterations, then apply LR
Foundations of Constraint Processing, Spring 2009
Local Search for CSPs
21
Big picture
• Agents do not communicate but share a
common context
• Agents keep kicking each other out of their
comfortable positions until every one is happy
• Charecterization:
[Hui Zou, 2003]
– Amazingly effective in solving very tight but solvable
instances
– Unstable in over-constrained cases
• Agents keep kicking each other out (livelock)
• Livelocks may be exploited to identify bottlenecks
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
22
ERA performance
70
Spring 2001b (B)
# agents in zero position
65
Fall 2001b
60
55
50
Spring 2003
45
40
Observation:
35
Fall 2002 (B)
30
25
20
iteration
15
1
20
39
58
77
96
115
191
172
153
134
Solvable vs. unsolvable instances:
ERA performance on solvable instances
# agents in zero position
45
• stable on solvable instances
• oscillates on unsolvable cases
Spring 2001b (O)
40
35
30
25
20
15
Fall 2002 (O)
iteration
10
1
20
39
58
77
96
115
134
153
172
191
ERA performance on unsolvable instances
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
23
Agent’s movement
Motion of agents
variable
40
• variable
• stable
• constant
20
0
index of position
1
51
101
151
201
251
301
351
401
451
stable
20
Observations:
10
0
1
51
101
151
201
251
301
351
401
Solvable
451
constant
Variable
None
Most
Stable
A few
A few
Constant
Most
None
30
20
10
Unsolvable
0
1
51
101
151
201
251
301
351
401
451
iteration
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
24
Outline
•
•
•
•
General principle
Main types: greedy & stochastic
When nothing works…
Evaluation methods
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
25
Random restart
• Principle
– When no progress is made, restart from a new
randomly selected state
– Save best results found so far (anytime algorithm)
• Repeat random restart
– For a fixed number of iterations
– Until best results have not been improved on for a
certain number of iterations. E.g., Geometric law
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
26
Outline
•
•
•
•
General principle
Main types: greedy & stochastic
When nothing works…
Evaluation methods
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
27
Evaluation: empirical
• Test on
– a given problem instance
– an ensemble of problem instances
(representative of a population)
• Experiment
– Run the algorithm thousands of times
– Measure some metric as a random variable
(e.g., the time needed to find a solution)
Foundations of Constraint Processing, Spring 2009
November
2, 2005
April 15, 2009
Local Search for CSPs
28
Comparing techniques
• Provide
– The probability distribution function (approximated by
the number of runs that took a given time to find a
solution)
– The cumulative distribution function (approximated by
the number of runs that found a solution within a
given time)
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
29
Comparing techniques
• Compare algorithms’ performance using
statistical tests (for confidence levels)
– t-test: assumes normal distribution of the measured metrics
– Nonparametric tests do not. Some match pairs, some do not.
• Consult/work with a statistician
1000
100.00%
Comparing
distributions
800
700
80.00%
Method1
Method2
60.00%
600
Frequency
Cumulative
Distribution
curves
Cumulative frequency
900
Method1
Method2
500
40.00%
400
300
20.00%
200
100
.00%
0
0
-1
1
3
5
7
9
11
13
2
4
6
8
10
12
CPU time
CPU time
Foundations of Constraint Processing, Spring 2009
April 15, 2009
Local Search for CSPs
30